JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(f\left( x \right) = {\sin ^4}\,x + {\cos ^4}\,x\). Then \(f\) is an increasing function in the interval
- A \(\left[ {\frac{{5\pi }}{8},\frac{{3\pi }}{4}} \right]\)
- B \(\left[ {\frac{\pi }{2},\frac{{5\pi }}{8}} \right]\)
- C \(\left[ {\frac{{\pi }}{4},\frac{{\pi }}{2}} \right]\)
- D \(\left[ {0,\frac{\pi }{4}} \right]\)
Answer & Solution
Correct Answer
(C) \(\left[ {\frac{{\pi }}{4},\frac{{\pi }}{2}} \right]\)
Step-by-step Solution
Detailed explanation
\(f(x)=\sin ^{4} x+\cos ^{4} x\) \(f^{\prime}(x)=4 \sin ^{3} x \cos x+4 \cos ^{3} x(-\sin x)\) \(=4 \sin x \cos x\left(\sin ^{2} x-\cos ^{2} x\right)\) \(=-2 \sin 2 x \cos 2 x\) \(=-\sin 4 x\) \(f(x)\) is increasing when \(f(x)>0\) \(\Rightarrow-\sin 4 x>0\)…
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